Representation theory of GL(n) overnon-Archimedean local fields

Dipendra Prasad A. Raghuram

March 29, 2007

Contents

1 Introduction 1

2 Generalities on representations 2

3 Preliminaries on GLn(F ) 6

4 Parabolic induction 11

5 Jacquet functors 15

6 Supercuspidal representations 20

7 Discrete series representations 26

8 Langlands classification 28

9 Certain classes of representations 309.1 Generic representations . . . . . . . . . . . . . . . . . . . . . . 309.2 Tempered representations . . . . . . . . . . . . . . . . . . . . 329.3 Unramified or spherical representations . . . . . . . . . . . . . 339.4 Iwahori spherical representations . . . . . . . . . . . . . . . . 35

10 Representations of local Galois Groups 38

11 The local Langlands conjecture for GLn 40

1

1 Introduction

The aim of these notes is to give an elementary introduction to representationtheory of p-adic groups via the basic example of GL(n). We have emphasisedthose topics which are relevant to the theory of automorphic forms. To keepthese notes to a reasonable size, we have omitted many proofs which wouldhave required a lot more preparation. We have given references throughoutthe notes either to original or more authoritative sources. Most of the omittedproofs can be found in the fundamental papers of Bernstein-Zelevinsky [2],Bushnell-Kutzko [5], or Casselman’s unpublished notes [7] which any seriousstudent of the subject will have to refer sooner or later.

Representation theory of p-adic groups is a very active area of reasearch.One of the main driving forces in the development of the subject is theLanglands program. One aspect of this program, called the local Langlandscorrspondence, implies a very intimate connection between representationtheory of p-adic groups with the representation theory of the Galois groupof the p-adic field. This conjecture of Langlands has recently been provedby Harris and Taylor and also by Henniart for the case of GL(n). We havegiven an introduction to Langlands’ conjecture for GL(n) as well as somerepresentation theory of Galois groups in the last two sections of these notes.

These notes are based on the lectures given by the first author in theworkshop ‘Automorphic Forms on GL(n)’ organized by Professors G. Harderand M.S. Raghunathan at ICTP, Trieste in August 2000. The authors wouldlike to thank the organizers for inviting them for this nice conference; theEuropean commission for financial support; Professor L. Gottsche for havingtaken care of all the details of our stay as well as the conference so well.

2 Generalities on representations

Let G denote a locally compact totally disconnected topological group. Inthe terminology of [2] such a group is called an l-group. For these notes thefundamental example of an l-group is the group GLn(F ) of invertible n× nmatrices with entries in a non-Archimedean local field F.

By a smooth representation (π, V ) of G we mean a group homomorphismof G into the group of automorphisms of a complex vector space V such thatfor every vector v ∈ V, the stabilizer of v in G, given by stabG(v) = {g ∈ G :π(g)v = v}, is open. The space V is called the representation space of π.

2

By an admissible representation (π, V ) of G we mean a smooth represen-tation (π, V ) such that for any open compact subgroup K of G, the invariantsin V under K, denoted V K , is finite dimensional.

Given a smooth representation (π, V ) of G, a subspace W of V is said tobe stable or invariant under G if for every w ∈ W and every g ∈ G we haveπ(g)w ∈ W. A smooth representation (π, V ) of G is said to be irreducible ifthe only G stable subspaces of V are (0) and V.

If (π, V ) and (π′, V ′) are two (smooth) representations of G then byHomG(π, π′) we denote the space of G intertwining operators from V intoV ′, i.e., HomG(π, π′) is the space of all linear maps f : V → V ′ such thatf(π(g)v) = π′(g)f(v) for all v ∈ V and all g ∈ G.

Let (π, V ) be a smooth representation of G. Let V ∗ denote the space oflinear functionals on V, i.e., V ∗ = HomC(V,C). There is an obvious actionπ∗ of G on V ∗ given by (π∗(g)f)(v) = f(π(g−1)v) for g ∈ G, v ∈ V andf ∈ V ∗. However this representation is in general not smooth. Let V ∨

denote the subspace of V ∗ which consists of those linear functionals in V ∗

whose stabilizers are open in G under the above action. It is easily seen thatV ∨ is stabilized by G and this representation denoted (π∨, V ∨) is called thecontragredient representation of (π, V ).

The following is a basic lemma in representation theory.

Lemma 2.1 (Schur’s Lemma) Let (π, V ) be a smooth irreducible admis-sible representation of an l-group G. Then the dimension of HomG(π, π) isone.

Proof : Let A ∈ HomG(π, π). Let K be any open compact subgroup of G.Then A takes the space of K-fixed vectors V K to itself. Choose a K suchthat V K 6= (0). Since V K is a finite dimensional complex vector space, thereexists an eigenvector for the action of A on V K , say, 0 6= v ∈ V K withAv = λv. It follows that the kernel of (A − λ1V ) is a nonzero G invariantsubspace of V. Irreducibility of the representation implies that A = λ1V . 2

Corollary 2.2 Any irreducible admissible representation of an abelian l-group is one dimensional.

Corollary 2.3 On an irreducible admissible representation (π, V ) of an l-group G the centre Z of G , operates via a character ωπ, i.e., for all z ∈ Zwe have π(z) = ωπ(z)1V . (This character ωπ is called the central characterof π.)

3

Exercise 2.4 (Dixmier’s lemma) Let V be a complex vector space ofcountable dimension. Let Λ be a collection of endomorphisms of V act-ing irreducibly on V. Let T be an endomorphism of V which commutes withevery element of Λ. Then prove that T acts as a scalar on V. (Hint. Think ofV as a module over the field of rational functions in one variable C(X) withX acting on V via T and use the fact that

{1

X−λ|λ ∈ C

}is an uncountable

set consisting of linearly independent elements in the C-vector space C(X).)Deduce that if G is an l-group which is a countable union of compact setsand (π, V ) is an irreducible smooth representation of G then the dimensionof HomG(π, π) is one.

One of the most basic ways of constructing representations is by the pro-cess of induction. Before we describe this process we need a small digressionon Haar measures.

Let G be an l-group. Let dlx be a left Haar measure on G. So in particulardl(ax) = dl(x) for all a, x ∈ G. For any g ∈ G the measure dl(xg) (where x isthe variable) is again a left Haar measure. By uniqueness of Haar measuresthere is a positive real number ∆G(g) such that dl(xg) = ∆G(g)dl(x). It iseasily checked that g 7→ ∆G(g) is a continuous group homomorphism andis called the modular character of G. Further dr(x) := ∆G(x)−1dl(x) is aright Haar measure. A left Haar measure is right invariant if and only if themodular character is trivial and in this case G is said to be unimodular.

Example 2.5 Let F be a non-Archimedean local field. The group G =GLn(F ) is unimodular. (In general a reductive p-adic group is unimodular.)This may be seen as follows. Note that ∆G is trivial on [G,G] as the range of∆G is abelian. Further by the defining relation ∆G is trivial on the centre Z.(Both these remarks are true for any group.) Hence ∆G is trivial on Z[G,G].Observe that Z[G,G] is of finite index in G and since positive reals admitsno non-trivial finite subgroups we get that ∆G is trivial.

Exercise 2.6 Let B be the subgroup of all upper triangular matrices inG = GLn(F ). Then show that B is not unimodular as follows. Let d∗y be aHaar measure on F ∗ and let dx be a Haar measure on F+. The normalizedabsolute value on F ∗ is defined by d(ax) = |a|Fdx for all a ∈ F ∗ and x ∈ F.

4

For b ∈ B given by

b =

y1

y2

.yn

11 xi,j

.1

,

show that a left Haar measure on B is given by the formula :

db =∏

i

d∗yi

∏1≤i<j≤n

dxi,j.

Deduce that the modular character ∆B of B is given by :

∆B(diag(a1, ..., an)) = |a1|1−nF |a2|3−n

F ...|an|n−1F .

Let δB = ∆−1B . Show that

δB(b) = |det(Ad(b)|Lie(U))|F

where Ad(b)|Lie(U) is the adjoint representation of B on the Lie algebra Lie(U)of U which is the unipotent radical of B. (In some places δB is used as thedefinition of the modular character, see for e.g., [1].)

For any l-group G, let ∆G denote the modular character of G. Let B bea closed subgroup of G. Let ∆ = ∆G|B ·∆−1

B . Let (σ,W ) be a smooth repre-sentation of B. To this is associated the (normalized) induced representationIndG

B(σ) whose representation space is :{f : G→ W

∣∣∣∣ (1) f(bg) = ∆(b)1/2σ(b)f(g), ∀b ∈ B, ∀g ∈ G(2) f(hu) = f(h) for all u ∈ Uf an open set in G

}.

The group G acts on this space by right translation, i.e., given x ∈ G andf ∈ IndG

B(σ) we have (x · f)(g) = f(gx) for all g ∈ G.We remark here that throughout these notes we deal with normalized

induction. By normalizing we mean the ∆1/2 factor appearing in (1) above.This is done so that unitary representations are taken to unitary representa-tions under induction. This simplifies some formulae like that which describesthe contragredient of an induced representation and complicates some otherformulae like Frobenius reciprocity.

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This induced representation admits a subrepresentation which is also an-other ‘induced’ representation called that obtained by compact induction.This is denoted as indG

B(σ) whose representation space is given by

indGB(σ) = {f ∈ IndG

B(σ) : f is compactly supported modulo B}.

The following theorem summarises the basic properties of induced repre-sentations.

Theorem 2.7 Let G be an l-group and let B be a closed subgroup of G.

1. Both IndGB and indG

B are exact functors from the category of smoothrepresentations of B to the category of smooth representations of G.

2. Both the induction functors are transitive,i.e., if C is a closed subgroupof B then IndG

B(IndBC) = IndG

C . A similar relation holds for compactinduction.

3. Let σ be a smooth representation of B. Then IndGB(σ)∨ = indG

B(σ∨).

4. (Frobenius Reciprocity) Let π be a smooth representation of G and σbe a smooth representation of B.

(a) HomG(π, IndGB(σ)) = HomB(π|B,∆1/2σ).

(b) HomG(indGB(σ), π) = HomB(∆−1/2σ, (π∨|B)∨).

Both the above identifications are functorial in π and σ.

The reader may refer paragraphs 2.25, 2.28 and 2.29 of [2] for a proof ofthe above theorem.

We recall some basic facts now about taking invariants under open com-pact subgroups. Let C be an open compact subgroup of an l-group G. LetH(G,C) be the space of compactly supported bi-C-invariant functions on G.This forms an algebra under convolution which is given by :

(f ∗ g)(x) =

∫G

f(xy−1)g(y) dy

with the identity element being vol(C)−1eC where eC is the characteristicfunction of C as a subset of G.

Let (π, V ) be a smooth admissible representation of G then the spaceof invariants V C is a finite dimensional module for H(G,C). The followingproposition is not difficult to prove. The reader is urged to try and fix aproof of this. (See paragraph 2.10 of [2].)

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Proposition 2.8 Let the notations be as above.

1. The functor (π, V ) 7→ (πC , V C) is an exact additive functor from thecategory of smooth admissible representations of G to the category offinite dimensional modules of the algebra H(G,C).

2. Let (π, V ) be an admissible representation of G such that V C 6= (0).Then (π, V ) is irreducible if and only if (πC , V C) is an irreducibleH(G,C) module.

3. Let (π1, V1) and (π2, V2) be two irreducible admissible representationsof G such that both V C

1 and V C2 are non-zero. We have π1 ' π2 as G

representations if and only if πC1 ' πC

2 as H(G,C) modules.

3 Preliminaries on GLn(F )

In this section we begin the study of the group G = GLn(F ) and its represen-tations. (Unless otherwise mentioned, from now on G will denote the groupGLn(F ).) We begin by describing this group and some of its subgroups whichare relevant for these notes. The emphasis is on various decompositions ofG with respect to these subgroups.

Let F be a non-Archimedean local field. Let OF be its ring of integerswhose unique maximal ideal is PF . Let $F be a uniformizer for F, i.e.,PF = $FOF . Let qF be the cardinality of the residue field kF = OF/PF .

The group G = GLn(F ) is the group of all invertible n × n matriceswith entries in F. A more invariant description is that it is the group of allinvertible linear transformations of an n dimensional F vector space V andin this case it is denoted GL(V ). If we fix a basis of V then we can identifyGL(V ) with GLn(F ).

We let B denote the subgroup of G consisting of all upper triangularmatrices. Let T denote the subgroup of all diagonal matrices and let Udenote the subgroup of all upper triangular unipotent matrices. Note thatT normalizes U and B is the semi-direct product TU. This B is called thestandard Borel subgroup with U being its unipotent radical and T is calledthe diagonal torus. We let U− to be the subgroup of all lower triangularunipotent matrices.

Let W , called the Weyl group of G, denote the group NG(T )/T whereNG(T ) is the normalizer inG of the torus T. It is an easy exercise to check that

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NG(T ) is the subgroup of all monomial matrices and so W can be identifiedwith Sn the symmetric group on n letters. We will usually denote a diagonalmatrix in T as diag(a1, ..., an).

We now introduce the Borel subgroup and more generally the parabolicsubgroups in a more invariant manner. Let V be an n dimensional F vectorspace. Define a flag in V to be a strictly increasing sequence of subspaces

W• = {W0 ⊂ W1 ⊂ · · ·Wm = V }.

The subgroup of GL(V ) which stabilizes the flag W•, i.e., with the propertythat gWi = Wi for all i is called a parabolic subgroup of G associated to theflag W•.

If {v1, ..., vn} is a basis of V then the stabilizer of the flags of the form

W• = {(v1) ⊂ (v1, v2) ⊂ · · · (v1, v2, ..., vn) = V }

is called a Borel subgroup. It can be seen that GL(V ) operates transitivelyon the set of such flags, and hence the stabilizer of any two such (maximal)flags are conjugate under GL(V ).

IfW• = {W0 ⊂ W1 ⊂ · · ·Wm = V },

then inside the associated parabolic subgroup P , there exists the normalsubgroup N consisting of those elements which operate trivially on Wi+1/Wi

for 0 ≤ i ≤ m − 1. The subgroup N is called the unipotent radical of P . Itcan be seen that there is a semi direct product decomposition P = MN with

M =m−1∏i=0

GL(Wi+1/Wi).

The decomposition P = MN is called a Levi decomposition of P with N theunipotent radical, and M a Levi subgroup of P.

We now introduce some of the open compact subgroups of G which willbe relevant to us. We let K = GLn(OF ) denote the subgroup of elements Gwith entries in OF and whose determinant is a unit in OF . This is an opencompact subgroup of G. The following exercise contains most of the basicproperties of K.

Exercise 3.1 Let V be an n dimesional F vector space. Let L be a lattice inV, i.e., an OF submodule of rank n. Show that stabG(L) is an open compact

8

subgroup of G = GL(V ). If C is any open compact subgroup then show thatthere is a lattice L such that C ⊂ stabG(L). Deduce that up to conjugacy Kis the unique maximal open compact subgroup of GLn(F ).

For every integer m ≥ 1, the map OF → OF/PmF induces a map K →

GLn(OF/PmF ). The kernel of this map, denoted Km is called the principal

congruence subgroup of level m. We also define K0 to be K. For all m ≥ 1,we have

Km = {g ∈ GLn(OF ) : g − 1n ∈ $mF Mn(OF )} .

Note that Km is an open compact subgroup of G and gives a basis of neigh-bourhoods at the identity. Hence G is an l-group, i.e., a locally compacttotally disconnected topological group.

We are now in a position to state the main decomposition theorems whichwill be of use to us later in the study of representations of G.

Theorem 3.2 (Bruhat decomposition)

G =∐

w∈W

BwB =∐

w∈W

BwU.

Proof: This is an elementary exercise in basic linear algebra involving rowreduction (on the left) and column reduction (on the right) by elementaryoperations. Since the rank of an element in G is n we end up with a monomialmatrix with such operations and absorbing scalars into T ⊂ B we end upwith an element in W. We leave the details to the reader.

The disjointness of the union requires more work and is not obvious. Werefer the reader to Theorem 8.3.8 of [15]. 2

Theorem 3.3 (Cartan decomposition) Let A = {diag($m1F , ..., $mn

F ) :mi ∈ Z, m1 ≤ m2 ≤ · · · ≤ mn}.

G =∐a∈A

K · a ·K

Proof : Let g ∈ G. After fixing a basis for an n dimensional F vector spaceV we identify GL(V ) with GLn(F ). Let L be the standard lattice in Vcorresponding to this basis. Let L1 be the lattice g(L). Let r be the leastinteger such that $r

FL is contained in L1 and let L2 = $rFL.

9

The proof of the Cartan decomposition falls out from applying the struc-ture theory of finitely generated torsion modules over principal ideal domains.In our context we would apply this to the torsion OF module L1/L2. We urgethe reader to fill in the details. 2

Theorem 3.4 (Iwasawa decomposition)

G = K ·B

Proof : We sketch a proof for this decomposition for GL2(F ). The proof forGLn(F ) uses induction on n and the same matrix manipulations as in theGL2(F ) case is used in reducing from n to n− 1.

Assume that G = GL2(F ). Let g = ( a bc d ) ∈ G. If a = 0 then write g as

w(w−1g) ∈ K ·B. If a 6= 0 and if a−1c ∈ OF then

g =

(1 0ca−1 1

) (∗ ∗0 ∗

)∈ K ·B.

If a−1c /∈ OF then replace g by kg where

k =

(1 1 +$r

F

1 $rF

)∈ K.

We may choose r large enough such that the (modified) a and c satisfy theproperty a−1c ∈ OF which gets us to the previous case and we may proceedas before. 2

Theorem 3.5 (Iwahori factorization) For m ≥ 1,

Km = (Km ∩ U−) · (Km ∩ T ) · (Km ∩ U).

Proof : This is an easy exercise in row and column reduction. See paragraph3.11 of [2]. 2

We are now in a position to begin the study of representations of G =GLn(F ). The purpose of these notes is to give a detailed account of irreduciblerepresentations of G with emphasis on those topics which are relevant to thetheory of automorphic forms.

The first point to notice is that most representations we deal with areinfinite dimensional. We leave this as the following exercise.

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Exercise 3.6 1. Show that the derived group of GLn(F ) is SLn(F ), thesubgroup of determinant one elements. Any character χ of F ∗ givesa character g 7→ χ(det(g)) of G = GLn(F ) and every character of Glooks like this.

2. Show that a finite dimensional smooth irreducible representation of Gis one dimensional and hence is of the form g 7→ χ(det(g)).

Note that any representation π of G can be twisted by a character χdenoted as π ⊗ χ whose representation space is the same as π and is givenby (π ⊗ χ)(g) = χ(det(g))π(g). It is trivial to see that π is irreducible ifand only if π ⊗ χ is irreducible and that ωπ⊗χ = ωπχ

n as characters of thecentre Z ' F ∗. For any complex number s we will denote π(s) to be therepresentation π ⊗ | · |sF .

Even if one is primarily interested in irreducible representations, somenatural constructions such as parabolic induction defined in the next section,force us to also consider representations which are not irreducible. Howeverthey still would have certain finiteness properties. We end this section witha quick view into these finiteness statements.

Let (π, V ) be a smooth representation of G. We call π to be a finitelygenerated representation if there exist finitely many vectors v1, ..., vm in Vsuch that the smallest G invariant subspace of V containing these vectors isV itself. We say that π has finite length if there is a sequence of G invariantsubspaces

(0) = V0 ⊂ V1 ⊂ · · · ⊂ Vm = V

such that each succesive quotient Vi+1/Vi for 0 ≤ i ≤ m− 1 is an irreduciblerepresentation of G.

The following theorem is a non-trivial theorem with a fairly long historyand various mathematician’s work has gone into proving various parts of it(especially Harish-Chandra, R. Howe, H. Jacquet and I.N. Bernstein). (SeeTheorem 4.1 of [2]. The introduction of this same paper has some relevanthistory.)

Theorem 3.7 Let (π, V ) be a smooth representation of G = GLn(F ). Thenthe following are equivalent :

1. π has finite length.

2. π is admissible and is finitely generated.

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Exercise 3.8 1. Prove the above theorem for G = GL1(F ) = F ∗.

2. Show that the sum of all characters of F ∗ on which the uniformizer$F acts trivially is an admissible representation which is not of finitelength.

3. Let f be the characteristic function of O×F thought of as a subset of F ∗.

Let V be the representation of F ∗ generated by f inside the regularrepresentation of F ∗ on C∞

c (F ∗). Show that V gives an example of asmooth finitely generated representation which is not of finite length.

4 Parabolic induction

One important way to construct representations of G is by the process ofparabolic induction.

Let P = MN be a parabolic subgroup of G = GLn(F ). Recall that Pis the stabilizer of some flag and its unipotent radical is the subgroup whichacts trivially on all successive quotients. More concretely, for a partitionn = n1 +n2 + · · ·+nk of n, let P = P (n1, n2, ..., nk) be the standard parabolicsubgroup given by block upper triangular matrices :

P =

g1 ∗ ∗ ∗g2 ∗ ∗

. .gk

: gi ∈ GLni(F )

.

The unipotent radical of P is the block upper triangular unipotent matricesgiven by :

NP = N =

1n1 ∗ ∗ ∗1n2 ∗ ∗

. .1nk

and the Levi subgroup of P is the block diagonal subgroup :

MP = M =

g1

g2

.gk

: gi ∈ GLni(F )

'k∏

i=1

GLni(F ).

12

Let (ρ,W ) be a smooth representation of M. Since M is the quotientof P by N we can inflate ρ to a representation of P (sometimes referredto as ‘extending it trivially across N’) also denoted ρ. Now we can considerIndG

P (ρ). (See § 2.) We say that this representation is parabolically inducedfrom M to G. To recall, IndG

P (ρ) consists of all locally constant functionsf : G→ Vρ such that

f(pg) = δP (p)1/2ρ(p)f(g)

where δP (p) = |det(Ad(p)|Lie(N))|F . (See Exercise 2.6.) The following theo-rem contains the basic properties of parabolic induction.

Theorem 4.1 Let P = MN be a parabolic subgroup of G = GLn(F ). Let(ρ,W ) be a smooth representation of M.

1. The functor ρ 7→ IndGP (ρ) is an exact additive functor from the category

of smooth representations of M to the category of smooth representa-tions of G.

2. IndGP (ρ) = indG

P (ρ).

3. IndGP (ρ∨) ' IndG

P (ρ)∨.

4. If ρ is unitary then so is IndGP (ρ).

5. If ρ is admissible then so is IndGP (ρ).

6. If ρ is finitely generated then so is IndGP (ρ).

Proof : General properties of induction stated in Theorem 2.7 gives (1).Iwasawa decomposition (Theorem 3.4) implies (2). Theorem 2.7 and (2)imply (3).

For a proof of (4) observe that given an M invariant unitary structure(see Definition 7.1) on W we can cook up a G invariant unitary structure onIndG

P (ρ) by integrating functions against each other which is justified by (2).We urge the reader to fill in the details.

We now prove (5). It suffices to prove the space of vectors fixed byKm (forany m) is finite dimensional where Km is the principal congruence subgroupof level m.

Let f ∈ IndGP (ρ) which is fixed by Km. Note that the Iwasawa decomposi-

tion (Theorem 3.4) gives that P\G/Km is a finite set. Let g1, ..., gr be a set of

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representatives for this double coset decomposition. We may and shall choosethese elements to be in K. The function f is completely determined by itsvalues on the elements gi. Since we have f(mgik) = δ

1/2P (m)ρ(m)f(gi) for all

m ∈M and k ∈ Km we get that f(gi) is fixed by M ∩ (giKmg−1i ) = M ∩Km

which is simply the principal congruence subgroup of level m for M. Henceeach of the f(gi) takes values in a finite dimensional space by admissibilityof ρ which implies that IndG

P (ρ)Km is finite dimensional.Now for the proof of (6) we give the argument to show that the space of

locally constant functions on P\G is a finitely generated representation of G.The argument for general parabolically induced representations is similar.To prove that the space of locally constant functions on P\G is finitelygenerated, it is sufficient to treat the case when P is the Borel subgroup Bof the group of upper triangular matrices. (We do this because the essenceof the argument is already seen in the case of B\G. It is however true that asubrepresentation of a finitely generated representation of G is itself finitelygenerated. See Theorem 4.19 of [2].)

The proof of finite generation of the space of locally constant functionson B\G depends on the Iwahori factorization (Theorem 3.5). Recall that ifKm is a principal congruence subgroup of level m then we have

Km = (Km ∩ U−) · (Km ∩ T ) · (Km ∩ U).

We note that there are elements in T which shrink K−m = Km ∩ U− to

the identity. For example if we take the matrix

µ = diag(1, $F , $2F , ..., $

n−1F )

then the powers of µ have the property that they shrink K−m to the identity,

i.e., limi→∞µ−iK−

mµi = {1}.

Let χX denote the characteristic function of a subset X of a certainambient space. Look at the translates of χB·K−

mby the powers µ−i. This will

give us,µ−i · χBK−

m= χBK−

mµi = χBµ−iK−mµi

Therefore translating χBK−m

by µ−i, we get the characteristic function ofBC for arbitrarily small open compact subgroups C. These characteristicfunctions together with their G translates clearly span all the locally constantfunctions on B\G, completing the proof of (6). 2

We would like to emphasize that even if ρ is irreducible the representationIndG

P (ρ) is in general not irreducible. However the above theorem assures us,

14

using Theorem 3.7, that it is of finite length. In general, understanding whenthe induced representation is irreducible is an extremely important one andthis is well understood for G. We will return to this point in the sectionon Langlands classification (§ 8). One instance of this is given in the nextexample.

Example 4.2 (Principal series) Let χ1, χ2, ..., χn be n characters of F ∗.Let χ be the character of B associated to these characters, i.e.,

χ

a1 ∗ ∗ ∗a2 ∗ ∗

. .an

= χ1(a1)χ2(a2)...χn(an).

The representation π(χ) = π(χ1, ..., χn) of G obtained by parabolically in-ducing χ to G is called a principal series representation of G. It turns outthat π(χ) is reducible if and only if there exist i 6= j such that χi = χj| · |Fwhere | · |F is the normalized multiplicative absolute value on F. In particularif χ1, ..., χn are all unitary then π(χ) is a unitary irreducible representation.

5 Jacquet functors

Parabolic induction constructs representations of GLn(F ) from representa-tions of its Levi subgroups. There is a dual procedure, more precisely, anadjoint functor, which constructs representations of Levi subgroups fromrepresentations of GLn(F ). The importance and basic properties of this con-struction was noted by Jacquet for the GLn(F ) case, which was generalizedto all reductive groups by Harish-Chandra.

Definition 5.1 Let P = MN be the Levi decomposition of a parabolicsubgroup P of G = GLn(F ). For a smooth representation (ρ, V ) of P ,define ρN to be the largest quotient of ρ on which N operates trivially. Letρ(N) = V (N) = {n · v − v|n ∈ N, v ∈ V }. Then

ρN = V/V (N).

This ρN is called the Jacquet functor of ρ. If ρ is a smooth representation ofG then the Jacquet functor ρN of ρ is just that of ρ restricted to P.

15

It is easily seen that the Jacquet functor ρN is a representation for M.This is seen by noting that M normalizes N. There is an easy and importantcharacterization of this subspace V (N) and this is given in the followinglemma.

Lemma 5.2 For a smooth representation V of N , V (N) is exactly the spaceof vectors v ∈ V such that ∫

KN

n · v dn = 0,

where KN is an open compact subgroup of N and dn is a Haar measure onN . (The integral is actually a finite sum.)

Proof : The main property of N used in this lemma is that it is a unionof open compact subgroups. Clearly the integral of the vectors of the formn·v−v on an open compact subgroup of N containing n is zero. On the otherhand if the integral equals zero, then v ∈ V (N) as it reduces to a similarconclusion about finite groups, which is easy to see.

2

The following theorem contains most of the basic properties of Jacquetfunctors. The reader is urged to compare this with Theorem 4.1.

Theorem 5.3 Let (π, V ) be a smooth representation of G. Let P = MN bea parabolic subgroup of G. Then

1. The Jacquet functor V → VN is an exact additive functor from thecategory of smooth representations of G to the category of smooth rep-resentations of M.

2. (Transitivity) Let Q ⊂ P be standard parabolic subgroups of G withLevi decompositions P = MPNP and Q = MQNQ. Hence MQ ⊂ MP ,NP ⊂ NQ, and MQ(NQ ∩MP ) is a parabolic subgroup of MP with MQ

as a Levi subgroup and NQ ∩MP as the unipotent radical, and we have(πNP

)NQ∩MP' πNQ

.

3. (Frobenius Reciprocity) For a smooth representation σ of M,

HomG(π, IndGP (σ)) ' HomM(πN , σδ

1/2P ).

16

4. If π is finitely generated then so is πN .

5. If π is admissible then so is πN .

Proof : For the proof of (1) it suffices to prove that V1 ∩ V (N) = V1(N)which is clear from the previous lemma. We urge the reader to fix proofs of(2) and (3). For (2) note that NQ = NP (NQ ∩MP ). For (3) it is an exercisein modifying the proof of the usual Frobenius reciprocity while using thedefinitions of parabolic induction and Jacquet functors. An easy applicationof Iwasawa decomposition (Theorem 3.4) gives (4).

Statement (5) is originally due to Jacquet. We sketch an argument belowas it is important application of Iwahori factorization. Let P = P (n1, ..., nk)be a standard parabolic subgroup and let P = MN be its Levi decomposi-tion. The principal congruence subgroup Km of level m admits an Iwahorifactorization which looks like

Km = K−mK

0mK

+m

where K+m = Km ∩ N and K0

m = Km ∩ M and K−m = Km ∩ N− where

P− = MN− is the opposite parabolic subgroup obtained by simply takingtransposes of all elements in P.

Let A : V → VN be the canonical projection. Under this projection weshow that

A(V Km) = (VN)K0m .

Clearly, using admissiblity of π, this would prove admissiblity of πN becauseK0

m is the principal congruence subgroup of level m of M and these form abasis of neighbourhoods at the identity for M.

Since A is a P equivariant map, we get that A(V Km) ⊂ (VN)K0m . Let

v ∈ (VN)K0m . Choose v ∈ V such that A(v) = v. For any z ∈ ZM , the centre

of M, given by

z =

$m1

F 1n1

$m2F 1n2

.$mk

F 1nk

let t(z) = maxi,j|mi − mj|. Let v1 = z−1v. Then A(v1) = A(z−1v) =z−1A(v) = z−1v. Since z is central in M , A(v1) is also fixed by K0

m.

17

Choose t(z) � 0 such that zK−mz

−1 ⊂ stabG(v). Hence K−m will fix v1 =

z−1v. To summarize, v1 is fixed by K−m and A(v1) is fixed by K0

m and K+m.

(The latter because N acts trivially on VN .)Choose a Haar measure on G such that vol(Km ∩ P ) = 1. Let v2 =∫

Kmπ(k)v1 dk. It is easy to check that A(v2) = z−1v. The upshot is that

given v ∈ (VN)K0m there exists z ∈ ZM such that z−1v ∈ A(V Km).

Now if v1, ..., vr are any linearly independent vectors in (VN)K0m then there

exists z ∈ ZM such that z−1v1, ..., z−1v are in A(V Km) and are also linearly

independent. This implies that the dimension of (VN)K0m is bounded above

by that of A(V Km) and hence they must be equal. 2

We now compute the Jacquet functor of the principal series representa-tion introduced in Example 4.2. As a notational convenience for a smoothrepresentation π of finite length of any l-group H, we denote by πss the semi-simplification of π, i.e., if π = π0 ⊃ π1 ⊃ · · · ⊃ πn = {0} with each πi/πi+1

irreducible, then πss = ⊕n−1i=0 (πi/πi+1). By the Jordan-Holder theorem, πss is

independent of the filtration π = π0 ⊃ π1 ⊃ · · · ⊃ πn = {0}.

Theorem 5.4 (Jacquet functor for principal series) Let π be the prin-

cipal series representation π(χ) = IndGLn(F )B (χ). Then the Jacquet functor of

π with respect to the Borel subgroup B = TU is given as a module for T as

(πU)ss '∑w∈W

χwδ1/2B

where χw denotes the character of the torus obtained by twisting χ by theelement w in the Weyl group W , i.e χw(t) = χ(w(t)).

Proof : The representation space of π can be thought of as a certain spaceof “functions on G/B twisted by the character χ”; more precisely, π can bethought of as the space of locally constant functions on G/B with values ina sheaf Eχ obtained from the character χ of the Borel subgroup B.

If Y is a closed subspace of a topological space X “of the kind that weare considering here”, e.g. locally closed subspaces of the flag variety, thenthere is an exact sequence,

0 → C∞c (X − Y, Eχ|X−Y ) → C∞

c (X, Eχ) → C∞c (Y, Eχ|Y ) → 0.

It follows that Mackey’s theory (originally for finite groups) about re-striction of an induced representation to a subgroup holds good for p-adic

18

groups too. Hence using the Bruhat decomposition GL(n) =∐

w∈W BwB,and denoting B ∩ wBw−1 to be T · Uw, we have

(ResBIndGB(χ))ss =

∑w∈W

indBB∩wBw−1((χδ

1/2B )w)

=∑w∈W

indBT ·Uw

((χδ1/2B )w)

=∑w∈W

C∞c (U/Uw, (χδ

1/2B )w).

We now note that the largest quotient of C∞c (U/Uw) on which U operates

trivially is one dimensional (obtained by integrating a function with respectto a Haar measure on U/Uw) on which the action of the torus T is thesum of positive roots which are not in Uw which can be seen to be [δB ·(δB)−w]1/2. (Here δ−w

B stands for the w translate of δ−1B .) This implies that

the largest quotient of C∞c (U/Uw, (χδ

1/2B )w) on which U operates trivially is

the 1 dimensional T -module on which T operates by the character χwδ1/2B .

Hence, ([Ind

GLn(F )B (χ)

]U

)ss'

∑w∈W

χwδ1/2B .

2

Corollary 5.5 If HomGLn(F )(IndGLn(F )B (χ), Ind

GLn(F )B (χ′)) is nonzero then

χ′ = χw for some w in W , the Weyl group.

Proof : This is a simple consequence of the Frobenius reciprocity combinedwith the calculation of the Jacquet functor done above. 2

Example 5.6 Let π(χ) denote a principal series representation of GL2(F ).Then

1. If χ 6= χw, then π(χ)U ' χδ1/2B ⊕ χwδ

1/2B .

2. If χ = χw, then π(χ)U is a non-trivial extension of T -modules:

0 → χδ1/2B → π(χ)U → χδ

1/2B → 0.

Exercise 5.7 With notation as in the previous exercise, prove that a prin-cipal series representation of GL2(F ) induced from a unitary character isirreducible.

19

Exercise 5.8 Let G be an abelian group with characters χ1 and χ2. Provethat if χ1 6= χ2, then any exact sequence of G-modules,

0 → χ1 → V → χ2 → 0,

splits.

We end this section with a theorem about Jacquet modules for parabol-ically induced representations. This theorem is at the basis for consideringsupercuspidal representations (which are those representations for which allJacquet modules are trivial).

Theorem 5.9 Let P = MN be a parabolic subgroup of G = GLn(F ). Let σbe a smooth irreducible representation of M. Let (π, V ) be any ‘subquotient’of IndG

P (ρ). Then the Jacquet module πN of π is non-zero.

Proof : Note that if π is subrepresentation of the induced representation thenit easily follows from Frobenius reciprocity. Indeed, we have

(0) 6= HomG(π, IndGP (ρ)) = HomM(πN , δ

1/2P ρ)

which implies the πN 6= (0). In general if π is a subquotient then there isa ‘trick’ due to Harish-Chandra (see Corollary 7.2.2 in [7]) using which πcan be realized as a subrepresentation of a Weyl group ‘twist’ of the inducedrepresentation, which will bring us to the above case. Since this trick is veryimportant and makes its presence in quite a few arguments we delineate itas the following theorem. 2

We will need a little bit of notation before we can state this theorem. LetP = MN be a parabolic subgroup of G. Let x ∈ NG(M) - the normalizer inG of M. Then we can consider the representation xρ of M whose representa-tion space is same as that of ρ and the action is given by xρ(m) = ρ(x−1mx).Now we can consider the induced representation IndG

P (xρ). Note that this rep-resentation is in general not equivalent to the original induced representation.

Theorem 5.10 Let P = MN be a parabolic subgroup of G = GLn(F ). Letσ be an smooth irreducible representation of M. Let (π, V ) be any irreduciblesubquotient of IndG

P (σ). Then there exists an element w ∈ W ∩NG(M) (whereW is the Weyl group) such that π is a subrepresentation of IndG

P (wσ).

20

6 Supercuspidal representations

A very important and novel feature of p-adic groups (compared to real re-ductive groups) is the existence of supercuspidal representations. We willsee that these representations are the building blocks of all irreducible ad-missible representations of p-adic groups. A complete set of supercuspidalsfor GLn(F ) was constructed by Bushnell and Kutzko in their book [5]. Thelocal Langlands correspondence, proved by Harris and Taylor and also byHenniart [10], interprets supercuspidal representations of GLn(F ) in termsof irreducible n dimensional representations of the Galois group of F. Beforewe come to the definition of a supercuspidal representation, we need to definethe notion of a matrix coefficient of a representation.

For a smooth representation (π, V ) ofGLn(F ) recall that (π∨, V ∨) denotesthe contragredient representation of (π, V ). For vectors v in π and v∨ in π∨,define the matrix coefficient fv,v∨ to be the function on GLn(F ) given byfv,v∨(g) = 〈v∨, π(g)v〉.

Theorem 6.1 Let (π, V ) be an irreducible admissible representation of G =GLn(F ). Then the following are equivalent :

1. One matrix coefficient of π is compactly supported modulo the centre.

2. Every matrix coefficient of π is compactly supported modulo the centre.

3. The Jacquet functors of π (for all proper parabolic subgroups) are zero.

4. The representation π does not occur as a subquotient of any represen-tation parabolically induced from any proper parabolic subgroup.

A representation satisfying any one of the above conditions of the theoremis called a supercuspidal representation. We refer the reader to paragraph 3.21of [2] for a proof of this theorem.

We note that every irreducible admissible representation of GL1(F ) = F ∗

is supercuspidal. However for GLn(F ) with n ≥ 2 since parabolic inductionis of no use in constructing supercuspidal representation, a totally new ap-proach is needed. One way to construct supercuspidal representations is viainduction from certain finite dimensional representations of compact opensubgroups. This has been a big program in recent times which has beencompleted for the case of GLn by Bushnell and Kutzko in their book [5].They prove that any supercuspidal representation of GLn(F ) is obtained by

21

induction from a finite dimensional representation of certain open compactmodulo centre subgroup of GLn(F ).

Here is a sample of such a construction in the simplest possible situa-tion. These are what are called depth zero supercuspidal representations ofGLn(F ). By a cuspidal representation of the finite groupGLn(Fq) we mean anirreducible representation for which all the Jacquet functors are zero (equiv-alently there are no non-zero vectors fixed by N(Fq) for unipotent radicalsN of any proper parabolic subgroup). Cuspidal representations of GLn(Fq)are completely known by the work of J.Green [9] in the 50’s.

Theorem 6.2 Consider a representation of GLn(Fq) to be a representationof GLn(OF ) via the natural surjection from GLn(OF ) to GLn(Fq). If σ is anirreducible cuspidal representation of GLn(Fq) thought of as a representationof GLn(OF ) and χ is a character of F ∗ whose restriction to O×

F is the sameas the central character of σ then χ · σ is a representation of F ∗GLn(OF ).

Let π be the compactly induced representation indGLn(F )F ∗GLn(OF )(χ · σ). Then π is

an irreducible admissible supercuspidal representation of GLn(F ).

The proof we give is based on Proposition 1.5 in [6]. To begin with we needa general lemma which describes the restriction of an induced representationin one special context that we are interested in.

Lemma 6.3 Let H be an open compact-mod-centre subgroup of a unimodularl-group G. Let (σ,W ) be a smooth finite dimensional representation of H. Letπ = indG

H(σ) be the compact induction of σ to a representation π of G. Thenthe restriction of π to H is given by :

π|H =[indG

H(σ)]H'

⊕g∈H\G/H

IndHH∩g−1Hg(

gσ|H∩g−1Hg)

where gσ is the representation of g−1Hg given by gσ(g−1hg) = σ(h) for allh ∈ H.

Proof of Lemma 6.3 : For g ∈ G, let Vg = C∞(HgH, σ) denote the spaceof smooth functions f on HgH such that f(h1gh2) = σ(h1)f(gh2). SinceHgH is open in G and is also compact mod H we get a canonical injectionVg → indG

H(σ) given by extending functions by zero outside HgH. All theinclusions Vg → π gives us a canonical map⊕

g∈H\G/H

Vg → π.

22

This map is an isomorphism of H modules. This can be seen as follows : Thismap is clearly injective. (Consider supports of the functions.) Surjectivityfollows from the definition of π. Note also that Vg is H stable if H acts byright shifts and the map Vg → π is H equivariant.

The map sending f → f from Vg to IndHH∩g−1Hg(

gσ|H∩g−1Hg) where f(h) =f(gh) is easily checked to be a well-defined H equivariant bijection. Thisproves the lemma. 2

Proof of Theorem 6.2 : Let H denote the subgroup F ∗GLn(OF ) which is anopen and compact-mod-centre subgroup of G = GLn(F ). For brevity let σalso denote the representation χ ·σ of H. We will prove irreducibility, admis-sibility and supercuspidality of π separately, which will prove the theorem.Irreducibility: We begin by proving irreducibility of π. We make the fol-lowing claim :Claim : For all g ∈ G−H, we have HomH∩g−1Hg(σ,

gσ) = (0).Assuming the claim for the time being we prove the theorem as follows.

The claim and the previous lemma implies that σ appears in π with multi-plicity one, i.e., dim(HomH(σ, π|H)) = 1.

Now suppose π is not irreducible then there exists an exact sequence ofnon-trivial G modules :

0 −→ π1 −→ π −→ π2 −→ 0.

We note that sinceH is compact-mod-centre, any representation (like σ, π1, π2

and π) on which F ∗ acts by a character is necessarily semi-simple as an H-module. Since π = indG

Hσ it embeds in IndGH(σ) as a G module and hence so

does π1. Using Frobenius reciprocity 2.7.4(a) we get that HomH(π1, σ) 6= (0),i.e., σ occurs in π1. By Frobenius reciprocity 2.7.4(b)

HomG(indGHσ, π2) = HomH(σ, (π∨2 |H)∨).

Since H is an open subgroup ((π∨2 )H)∨ = π2 as H-modules. Thus we getthat σ occurs in π2 also but this contradicts the fact that σ occurs withmultiplicity one in π.

It suffices now to prove the claim. Since g ∈ G − H, we can assumewithout loss of generality, using the Cartan decomposition (see Theorem 3.3),that g = diag($m1

F , ..., $mn−1

F , 1) where m1 ≥ ... ≥ mn−1 ≥ 0. Choose ksuch that mk ≥ 1 and mk+1 = 0. Suppose that HomH∩g−1Hg(σ,

gσ) 6= (0).Then we also have HomNk(O)∩g−1Nk(O)g(σ,

gσ) 6= (0) where Nk is the unipotent

23

radical of the standard parabolic subgroup corresponding to the partitionn = k+(n−k). We have Nk(O)∩gNk(O)g−1 ⊂ Nk(P) - the notations beingobvious. Since σ is inflated from GLn(Fq),

gσ is trivial onNk(O)∩g−1Nk(O)g,i.e., HomNk(O)∩g−1Nk(O)g(σ,1) 6= (0). This contradicts the fact that (σ,W ) isa cuspidal representation of GLn(Fq). (We urge the reader to justify this laststatement.)

Admissibility: We now prove that π is an admissible representation. Itsuffices to show that for m ≥ 1, πKm is finite dimensional, where Km is theprincipal congruence subgroup of level m.

Note that πKm consists of locally constant functions f : G → W suchthat

f(hgk) = σ(h)f(g), ∀h ∈ H,∀g ∈ G, ∀k ∈ Km.

Using the Cartan decomposition (see Theorem 3.3), we may choose repre-sentatives for the double cosets H\G/Km which look like g = a · k, wherea = diag($m1

F , · · · , $mnF ) with 0 = m1 ≤ m2 ≤ · · · ≤ mn, and k ∈ K/Km.

For such an a, let t(a) = max{mi+1 − mi : 1 ≤ i ≤ n − 1}. Note thatif t(a) ≥ m, and if t(a) = mj+1 − mj, then Uj(O) ⊂ gKmg

−1 ∩ H whereUj is the unipotent radical of the maximal parabolic corresponding to thepartition n = j + (n− j). Therefore,

σ(u) · f(g) = f(ug) = f(g · g−1ug) = f(g),

i.e., f(g) is a vector in W which is fixed by Uj(O) and hence Uj(Fq). Cuspi-dality of (σ,W ) implies that f(g) = 0.

This shows that if 0 6= f ∈ πKm , then f can be supported only ondouble cosets HakKm with t(a) < m, which is a finite set, proving the finitedimensionality of πKm .

Supercuspidality: Let P = M ·N be a parabolic subgroup of G. To provethat π is supercuspidal, we need to show that the Jacquet module πN = (0).It suffices to show that (πN)∗ = (0). Note that HomN(π,C) ∼= (π∗)N is thespace of N -fixed vectors in the vector space dual of π. We will show that(π∗)N = 0.

Since π = indGH(σ), the dual vector space π∗ may be identified with

C∞(H\G, σ∗) which is the space of locally constant functions φ on G withvalues in W ∗ such that φ(hg) = σ∗(h)φ(g). (We urge the reader to justifythis identification; it boils down to saying that the dual of a direct sum of

24

vector spaces is the direct product of vector spaces.) Hence (π∗)N consistsof locally constant functions φ : G→ W ∗ such that

φ(hgn) = σ∗(h)φ(g), ∀h ∈ H, ∀g ∈ G, ∀n ∈ N.

Using Iwasawa decomposition, we may take representatives for double cosetsH\G/N to lie in M . For m ∈M , note that

N(O) = N ∩H = H ∩mNm−1.

Hence for all h ∈ N(O), we have

σ∗(h)φ(m) = φ(hm) = φ(m ·m−1hm) = φ(m),

i.e., φ(m) ∈ W ∗ is fixed by N(Fq). Cuspidality of (σ,W ) implies cuspidalityof (σ∗,W ∗) which gives that φ(m) = 0. Hence (π∗)N = (0). 2

Exercise 6.4 With notations as in Theorem 6.2 show that

indGLn(F )F ∗GLn(OF )(χ · σ) = Ind

GLn(F )F ∗GLn(OF )(χ · σ).

We next have the following basic theorem which justifies the assertionmade in the beginning of this section that supercuspidal representations arethe building blocks of all irreducible representations. This theorem will berefined quite a bit in the section on Langlands classification.

Theorem 6.5 Let π be an irreducible admissible representation of G =GLn(F ). Then there exists a Levi subgroup M and a supercuspidal repre-sentation ρ of M such that π is a subrepresentation of the representation ofG obtained from ρ by the process of parabolic induction.

Proof : Let P be a parabolic subgroup of GLn(F ) which is smallest for theproperty that the Jacquet functor with respect to P is non-zero. (So P = G isa possibility which occurs if and only if the representation is supercuspidal.)If P = MN , it is clear from theorem 5.3 that any irreducible subquotient ofπN is a supercuspidal representation of M . Let ρ be an irreducible quotient ofπN as an M module. Since HomM(πN , ρ) 6= (0), it follows from the Frobeniusreciprocity that

HomGLn(F )(π, IndGLn(F )P (δ

−1/2P ρ)) 6= (0),

25

which gives a realization of π inside a representation parabolically inducedfrom a supercuspidal representation. 2

Given the previous theorem one might ask if given an irreducible π ‘howmany’ induced representations can it occur in? The next theorem says that πoccurs in essentially only one such induced representation. Such a uniquenessassertion then allows us to talk of the supercuspidal support of the given rep-resentation. This is the following theorem due to Bernstein and Zelevinsky.(see Theorem 2.9 of [3].) We need some notation to state the theorem.

If π is a representation of finite length then let JH(π) denote the setof equivalence classes of irreducible subquotients of π. Further, let JH0(π)denote the set of all irreducible subquotients of π counted with multiplicity.So each ω in JH(π) is contained in JH0(π) with some multiplicity.

Theorem 6.6 Let P = MN and P ′ = M ′N ′ be standard parabolic subgroupsof G = GLn(F ). Let σ (resp. σ′) be an irreducible supercuspidal represen-tation of M (resp. M ′). Let π = IndG

P (σ) and π′ = IndGP ′(σ′). Then the

following are equivalent.

1. There exists w ∈ W such that wMw−1 = M ′ and wσ = σ′.

2. HomG(π, π′) 6= (0).

3. JH(π) ∩ JH(π′) is not empty.

4. JH0(π) = JH0(π′).

7 Discrete series representations

Note that one part of Theorem 6.1 says that every matrix coefficient of anirreducible supercuspidal representation is compactly supported modulo thecentre. In general analytic behaviour of matrix coefficients dictate propertiesof the representation. Now we consider a larger class of representations whichare said to be in the discrete series for G. These have a characterization interms of their matrix coefficients being square integrable modulo the centre.

We need a few definitions now. Albeit we have mentioned the notion ofa unitary representation before (see Theorem 4.1) we now give a definition.

Definition 7.1 Let (π, V ) be a smooth representation of G. We say π is aunitary representation of G if V has an inner product 〈 , 〉 (also called an

26

unitary structure) which is G invariant, i.e., 〈π(g)v, π(g)w〉 = 〈v, w〉 for allg ∈ G and all v, w ∈ V. In general, V equipped with 〈 , 〉 need only be apre-Hilbert space.

Definition 7.2 Let (π, V ) be a smooth irreducible representation of G. Wesay π is essentially square integrable if there is a character χ : F ∗ → R>0

such that |fv,v∨(g)|2χ(det g) is a function on Z\G for every matrix coefficientfv,v∨ of π, and ∫

Z\G|fv,v∨(g)|2χ(det g) dg <∞.

If χ can be taken to be trivial then π is said to be a square integrable repre-sentation and in this case it is said to be in the discrete series for G.

Exercise 7.3 1. Let ν be a unitary character of F ∗. Let L2(Z\G, ν) de-note the space of measurable functions f : G → C such that f(zg) =ν(z)f(g) for all z ∈ Z and g ∈ G and∫

Z\G|f(g)|2 dg <∞.

Show that L2(Z\G, ν) is a unitary representation of G where G actson these functions by right shifts.

2. Let (π, V ) be a discrete series representation. Check that its centralcharacter ωπ is a unitary character. Let 0 6= l ∈ V ∨. Show that themap v 7→ fv,l is a non-zero G equivariant map from π into L2(Z\G,ωπ).Hence deduce that π is a unitary representation which embeds as asubrepresentation of L2(Z\G,ωπ).

The classification of discrete series representations is due to Bernsteinand Zelevinsky. We summarize the results below. To begin with we need aresult on when a representation parabolically induced from a supercuspidalrepresentation is reducible. (See Theorem 4.2 of [3].)

Theorem 7.4 Let P = P (n1, ..., nk) = MN be a standard parabolic subgroupof G. Let σ = σ1 ⊗ · · · ⊗ σk be an irreducible representation of M with everyσi an irreducible supercuspidal representation of GLni

(F ). The parabolicallyinduced representation IndG

P (σ) is reducible if and only if there exist 1 ≤i, j ≤ k with i 6= j, ni = nj and σi ' σj(1) = σj| · |F .

27

We now consider some very specific reducible parabolically induced rep-resentations. Let n = ab and let σ be an irreducible supercuspidal repre-sentation of GLa(F ). Let P = MN be the standard parabolic subgroupcorresponding to n = a + a + ... + a the sum taken b times, so M =GLa(F )× ...×GLa(F ) the product taken b times. Let ∆ denote the segment

∆ = (σ, σ(1), ..., σ(b− 1))

thought of as the representation σ⊗ σ(1)⊗ ...⊗ σ(b− 1) of M. Let IndGP (∆)

denote the corresponding parabolically induced representation of G. By The-orem 7.4 this representation is reducible. We are now in a position to statethe following theorem. (This theorem, as recorded in Theorem 9.3 of Zelevin-sky’s paper [16], is normally attributed to Bernstein.)

Theorem 7.5 With notations as above, we have :

1. For any segment ∆ the induced representation IndGP (∆) has a unique

irreducible quotient. This irreducible quotient will be denoted Q(∆).

2. For any segment ∆ the representation Q(∆) is an essentially squareintegrable representation. Every essentially square integrable represen-tation of G is equivalent to some Q(∆) for a uniquely determined ∆,i.e., for a uniquely determined a, b and σ.

3. The representation Q(∆) is square integrable if and only if it is unitaryand also if and only if σ((b− 1)/2) is unitary.

Example 7.6 (Steinberg Representation) One important example of adiscrete series representation is obtained as follows. Take a = 1 and b =n with the notations as above. Hence the parabolic subgroup P is justthe standard Borel subgroup B. Let σ = | · |(1−n)/2

F as a (supercuspidal)representation of GLa(F ) = F ∗. Then the representation IndG

P (∆) is just theregular representation of G on smooth functions on B\G. The correspondingQ(∆) is called the Steinberg representation of G.We will denote the Steinbergrepresentation for GLn(F ) by Stn. It is a square integrable representationwith trivial central character. It can also be defined as the alternating sum

Stn =∑B⊂P

(−1)rank(P )C∞c (P\G),

28

where B is a fixed Borel subgroup of G, P denotes a parabolic subgroup ofG containing B, and rank(P ) denotes the rank of [M,M ] where M is a Levisubgroup of P .

The essentially square integrable representations Q(∆) are also calledgeneralized Steinberg representations.

Exercise 7.7 Show that the Jacquet module with respect to B of the Stein-berg representation of GLn(F ) is one dimensional. What is the character ofT on this one dimensional space?

Exercise 7.8 Show that the trivial representation of GLn(F ) is not essen-tially square integrable.

8 Langlands classification

In this section we state the Langlands classification for all irreducible ad-missible representations of GLn(F ). The supercuspidal repsentations wereintroduced in § 6. Then we used supercuspidal representations to constructall the essentially square integrable representations (the Q(∆)’s) in § 7. Notevery representation is accounted for by this construction as for instance thetrivial representation is not in this set (unless n = 1!). The Langlands classi-fication builds every irreducible representation starting from the essentiallysquare integrable ones.

The theorem stated below is due to Zelevinsky. (See Theorem 6.1 of [16].)Before we can state the theorem we need the notion of when two segmentsare linked. Let ∆ = (σ, ..., σ(b − 1)) and ∆′ = (σ′, ..., σ′(b′ − 1)) be twosegments where σ (resp. σ′) is an irreducible supercuspidal representation ofGLa(F ) (resp. GLa′(F )). We say ∆ and ∆′ are linked if neither of them isincluded in the other and their union is a segment (so in particular a = a′).We say that ∆ precedes ∆′ if they are linked and there is a positive integerr such that σ′ = σ(r).

Theorem 8.1 (Langlands classification) For 1 ≤ i ≤ k, let ∆i be a seg-ment for GLni

(F ). Assume that for i < j, ∆i does not precede ∆j. Letn = n1 + ... + nk. Let G = GLn(F ) and let P be the parabolic subgroupof G corresponding to this partition. Then :

29

1. The parabolically induced representation IndGP (Q(∆1)⊗···⊗Q(∆k)) ad-

mits a unique irreducible quotient which will be denoted Q(∆1, ...,∆k).

2. Any irreducible representation of G is equivalent to some Q(∆1, ...,∆k)as above.

3. If ∆′1, ...,∆

′k′ is another set of segments satisfying the hypothesis then we

have Q(∆1, ...,∆k) ' Q(∆′1, ...,∆

′k′) if and only if k = k′ and ∆i = ∆′

s(i)

for some permutation s of {1, ..., k}.

Exercise 8.2 Give a realization of the trivial representation of G as a rep-resentation Q(∆1, ...,∆k) in the Langlands classification.

Exercise 8.3 (Representations of GL2(F )) In this exercise all the irre-ducible admissible representations ofGL2(F ) are classified. LetG = GL2(F ).

1. Let χ be a character of F ∗. Then g 7→ χ(det(g)) gives a character ofG. Show that these are all the finite dimensional irreducible admissiblerepresentations of G.

2. Let χ be a character of G as above. Let St2 be the Steinberg repre-sentation of G and let St2(χ) = St2 ⊗ χ be the twist by χ. Show thatany principal series representation is either irreducible or is reduciblein which case, up to semi-simplification looks like χ⊕ St2(χ) for somecharacter χ.

3. Use the Langlands classification and make a list of all irreducible ad-missible representations of G. Show that any irreducible admissble rep-resentation is either one dimensional or is infinite dimensional and inwhich case it is either supercuspidal, or of the form St2(χ) or is anirreducible principal series representation π(χ1, χ2).

9 Certain classes of representations

In this section we examine certain important classes of representations of G =GLn(F ). These notions are especially important in the theory of automorphicforms. (For readers familiar with such terms, local components of globalrepresentations tend to have such properties.)

30

9.1 Generic representations

Recall our notation that U is the unipotent radical of the standard Borelsubgroup B = TU where T is the diagonal torus. (So U is the subgroup ofall upper triangular unipotent matrices.) Since T normalizes U we get anaction of T on the space U consisting of all characters of U. A maximal orbitof T on U is called a generic orbit and any character of U in a generic orbitis called a generic character of U. It is easily seen that a character of U isgeneric if and only if its stabilizer in T is the centre Z.

We can construct (generic) characters of U as follows. Let u = u(i, j)be any element in U. Let ψF be a non-trivial additive character of U. Leta = (a1, ..., an−1) be any (n − 1) tuple of elements of F. We get a characterof U by

u 7→ ψa(u) = ψF (a1u(1, 2) + a2u(2, 3) + ...+ an−1u(n− 1, n)).

The reader is urged to check that every character of U is one such ψa. Furtherit is easily seen that ψa is generic if and only if each ai is non-zero.

We will now fix one generic character of U , denoted Ψ. Fix for once and forall a non-trivial additive character ψF of F such that the maximal fractionalideal of F on which ψF is trivial is OF . We fix the following generic characterof U given by:

u 7→ Ψ(u) = ψF (u(1, 2) + ...+ u(n− 1, n)).

Now consider the representation IndGU (Ψ) which is the induction from

U to G of the character Ψ. Note that the representation space of IndGU (Ψ)

consists of all smooth functions f on G such that f(ng) = Ψ(n)f(g) for alln ∈ U and g ∈ G.

Definition 9.1 An irreducible admissible representation (π, V ) is said to begeneric if HomG(π, IndG

U(Ψ)) is non-zero. If π is generic then using Frobeniusreciprocity we get that there is a non-zero linear functional ` : V → C suchthat `(π(n)v) = Ψ(n)`(v) for all v ∈ V and n ∈ U. Such a linear functionalis called a Whittaker functional for π. If π is generic then the representationspace V may be realized on a certain space of functions f with the propertythat f(ng) = Ψ(n)f(g) for all n ∈ U and g ∈ G and the action of G on Vis by right shifts on this realization. Such a realization is called a Whittakermodel for π.

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One of the main results on generic representations is the following theorem(due to Shalika [14]) which says that Whittaker models are unique when theyexist. This theorem is important for the global theory, for instance, to provemultiplicity one for automorphic forms on GL(n). Also, one way to attachlocal Euler factors for generic representations of G is via Whittaker models.

Theorem 9.2 (Multiplicity one for Whittaker models) Let (π, V ) bean irreducible admissible representation of GLn(F ). Then the dimension ofthe space of Whittaker functionals is at most one, i.e.,

dimC(HomG(π, IndGU (Ψ))) ≤ 1.

Put differently, if π admits a Whittaker model then it admits a unique one.

The question now arises as to which representations are actually generic.The first theorem in this direction was due to Gelfand and Kazhdan whichsays that every irreducible admissible supercuspidal representation is generic.The classification of all generic representations of GLn(F ) is a theorem due toZelevinsky (see Theorem 9.7 of [16]) which states that π is generic if and onlyif it is irreducibly induced from essentially square integrable representations.In particular one has that discrete series representations are always generic.We would like to point out that this phenomenon is peculiar to GLn(F ) andis false for general reductive p-adic groups.

Theorem 9.3 (Generic representations) Let π = Q(∆1, ...,∆k) be anirreducible admissible representation of GLn(F ). Then π is generic if andonly if no two of the ∆i are linked in which case IndG

P (Q(∆1)⊗ · · · ⊗Q(∆k))is an irreducible representation, and

π == Q(∆1, ...,∆k) = IndGP (Q(∆1)⊗ · · · ⊗Q(∆k)).

Exercise 9.4 1. Use the definition of genericity to show that one dimen-sional representations are never generic. Now verify the above theoremfor one dimensional representations.

2. Let G = GL2(F ). Show that an irreducible admissible representationis generic if and only if it is infinite dimensional.

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9.2 Tempered representations

The notion of a tempered representation is important in the global theoryof automorphic forms. The importance is evidenced by what is called thegeneralized Ramanujan conjecture which says that every local component ofa cuspidal automorphic representation of GL(n) is tempered. In this sectionwe give the definition of temperedness and state a thereom due to Jacquet[12] which says when a representation is tempered. (It is closely related toZelevinsky’s theorem in the previous section.) It says that a representationwith unitary central character is tempered if and only if it is irreduciblyinduced from a discrete series representation (as against essentially squareintegrable representations to get the generic ones).

Definition 9.5 An irreducible admissible representation (π, V ) is temperedif ωπ the central character of π is unitary and if one (and equivalently every)matrix coefficient fv,v∨ is in L2+ε(Z\G) for every ε > 0.

Every unitary supercuspidal representation of G is tempered, since matrixcoefficients of supercuspidals are compactly supported modulo centre andhence in L2+ε(Z\G). One may drop the unitarity and rephrase this as everysupercuspidal being essentially tempered with an obvious meaning given tothe latter.

Theorem 9.6 (Tempered representations) Let π = Q(∆1, ...,∆k) be anirreducible admissible representation of GLn(F ). Then π is tempered if andonly if one actually has

π = IndGP (Q(∆1)⊗ · · · ⊗Q(∆k))

with every Q(∆i) being square-integrable.

From Theorem 9.3 and Theorem 9.6 we get that every tempered rep-resentation of GLn(F ) is generic. We emphasize this point in light of theglobal context. A theorem of Shalika [14] says that every local componentof a cuspidal automorphic representation of GLn is generic. The generalizedRamanujan conjecture for GLn says that every such local component is tem-pered. The reader is urged to construct examples of generic representationsof GLn(F ) which are not tempered.

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9.3 Unramified or spherical representations

The notion of unramified representations is again of central importance inthe global theory as almost all local components of global representations areunramified. An unramified representation is a GLn analogue of an unramifiedcharacter of F ∗ which is just a character trivial on the units O×

F . Such acharacter is called unramified because every character corresponding to anunramified extension via local class field theory is unramified (the norm mapbeing surjective on the units).

Recall that K = GLn(OF ) is the (up to conjugacy) unique maximal com-pact subgroup of G = GLn(F ). We let HK denote the spherical Hecke algebraof G which is the space of compactly supported bi-K-invariant functions onG. This is an algebra under convolution

(f ∗ g)(x) =

∫G

f(xy−1)g(y) dy

where dy is a Haar measure on G normalized such that vol(K) = 1. Theidentity element in HK is just the characteristic function of K.

Exercise 9.7 (Gelfand) Show that HK is a commutative algebra. Con-sider the transpose map on G to show that it induces a map on HK whichis both an involution and an anti-involution. (Hint: Use Thereom 3.3.)

Actually more is known about this Hecke algebra and we state the fol-lowing theorem of Satake regarding the structure of HK .

Theorem 9.8 (Satake Isomorphism) The spherical Hecke algebra HK iscanonically isomorphic to the Weyl group invariants of the space of Laurentpolynomials in n variables, i.e.,

HK ' C[t1, t−11 , ..., tn, t

−1n ]W .

Definition 9.9 Let (π, V ) be an irreducible admissible representation of G.It is said to be unramified if it has a non-zero vector fixed by K. The spaceof K fixed vectors denoted V K is a module for the spherical Hecke algebraHK .

Let (π, V ) be an irreducible unramified representation of G. We knowthat V K is a module for HK . We can use Proposition 2.8 and the fact that

34

HK is commutative to get that the space of fixed vectors is actually onedimensional and hence this gives a character of the spherical Hecke algebrawhich is called the spherical character associated to π. Using this propositionagain gives that the spherical character uniquely determines the unramifiedrepresentation.

We now construct all the unramified representations of G. Let χ1, ..., χn

be n unramified characters of F ∗ (i.e., they are all trivial on the units of F ).Let π = π(χ1, ..., χn) be the corresponding principal series representationof GLn(F ). It is an easy exercise using the Iwasawa decomposition to seethat π admits a non-zero vector fixed by K which is unique up to scalars.Hence π admits a unique subquotient which is unramified. It turns out thatif the characters χi are ordered to satisfy the ‘does not precede’ conditionof Theorem 8.1 then the unique irreducible quotient Q(χ1, ..., χn) of π isactually unramified and so is the unique unramified subquotient of π.

We now state the main theorem classifying all the unramified represen-tations of GLn(F ).

Theorem 9.10 (Spherical representations) Let χ1, ..., χn denote n un-ramified characters of F ∗ which are ordered to satisfy the ‘does not precede’condition of Theorem 8.1. Then the representation Q(χ1, ..., χn) is an un-ramified representation of GLn(F ). Further, every irreducible admissible un-ramified representation is equivalent to such a Q(χ1, ..., χn).

Proof : We briefly sketch the proof of the second assertion in the theorem.We show that the Q(χ1, ..., χn) exhaust all the unramified representations ofG. This is done by appealing to Proposition 2.8 and the Satake isomorphism.The proof boils down to showing that every character of the spherical Heckealgebra is accounted for by one of the characters coming from a Q(χ1, ..., χn)and so by appealing to Statement (3) of Proposition 2.8 we would be done.

Now via the Satake isomorphism, a character of HK determines and isdetermined by a set of n non-zero complex numbers which are the valuestaken by the character on the variables t1, ..., tn. Now it is easy to check thatthe spherical character of Q(χ1, ..., χn) (or π(χ1, ..., χn) which is more easierto work with) takes the value χi($F ) on the variable ti. The remark that anunramified character is completely determined by its value on a uniformizerwhich can be any non-zero complex number finishes the proof. 2

35

Example 9.11 Let π = Q(χ1, χ2) be an irreducible admissible unitary un-ramified generic representation of GL2(F ). Then for i = 1, 2 it can be shownthat

q−1/2 ≤ |χi($F )| ≤ q1/2.

In particular, there are such representations which are not tempered. TheRamanujan conjecture for GL2(F ) would assert that if π is a local componentof a cuspidal automorphic representation of GL2 then π is tempered. Notethat almost all such local components are unitary unramified and generic andthe Ramanujan conjecture would boil down to showing that |χi($F )| = 1.See the discussion on pp.332-334 of [1] for related matters and in particularas to how the Langlands program implies the Ramanujan conjecture.

9.4 Iwahori spherical representations

This is a class of representations which may not have an immediate motiva-tion from the global theory but is important all the same for various localreasons. (Although justifying this claim may take us outside the scope ofthese notes as it requires more knowledge of the structure theory of p-adicgroups.)

The Iwahori subgroup I of G = GLn(F ) that we will be looking at is theinverse image of the standard Borel subgroup under the canonical map fromGLn(OF ) → GLn(Fq). By definition, I is a subgroup of K.

We will be looking at those representations of G which have vectors fixedby I. For this reason, as before, we need to consider the Iwahori Hecke algebraHI which is the space of compactly supported bi-I-invariant functions on Gwith the algebra structure being given by convolution.

Definition 9.12 An irreducible admissible representation (π, V ) of G iscalled Iwahori spherical if it has a non-zero vector fixed by the Iwahori sub-group I. In this case the space of fixed vectors V I is a simple module denoted(πI , V I) for the Iwahori Hecke algebra HI .

To begin with, unlike the spherical case, the Iwahori Hecke algebra isnot commutative and admits simple modules which are not one dimensional.The structure of HI is also a little complicated to describe and would takeus outside the scope of these notes. (See § 3 of Iwahori-Matsumoto [11] forthe original description of this algebra. See also § 3 of Borel [4] for a morepleasanter-to-read version of the structure of HI .) We record the following

36

rather special lemma on representations of the Iwahori Hecke algebra. SeeProposition 3.6 of [4], although Borel deals only with semi-simple groups theproof goes through mutatis mutandis to our case. We need some notation.For any g ∈ G, let eIgI be the characteristic function of IgI as a subsetof G. Clearly eIgI ∈ HI . If we normalize the Haar measure on G such thatvol(I) = 1, then eI is the identity of element of HI .

Lemma 9.13 Let (σ,W ) be a finite dimensional representation of HI . Thenfor any g ∈ G the endomorphism σ(eIgI) is invertible.

The main fact about Iwahori spherical representations is the followingtheorem due to Borel and Casselman. (See Theorem 3.3.3. of [7] and astrengthening of it in Lemma 4.7 of [4].) Recall our notation that B = TUis the standard Borel subgroup of G. The Iwahori subgroup I admits an‘Iwahori-factorization’ with respect to B as

I = I−I0I+

where I− = I ∩ U−, I0 = I ∩ T which is the maximal compact subgroup ofT and I+ = I ∩ U.

Theorem 9.14 Let (π, V ) be an admissible representation of G. Then thecanonical projection from V to the Jacquet module VU induces an isomor-phism from V I onto (VU)I0

.

Proof : Let A : V → VU be the canonical projection map. By the proof ofstatement (5) of Theorem 5.3 we have that A(V I) = (VU)I0

. We now have toshow that A is injective on V I .

Suppose v ∈ V I is such that A(v) = 0. Then v ∈ V I ∩ V (U). Choosea compact open subgroup U1 of U such that v ∈ V (U1) which gives thatπ(eU1)v = 0. (For any compact subset C of G we let π(eC)v stand for∫

Cπ(g)v dg.)Choose r � 0 such that µ−rU1µ

r ⊂ I+ where µ = diag(1, $F , ..., $n−1F ).

We then get that π(eI+)(π(µ−r)v) = 0. The Iwahori factorization gives thatπ(eI)π(µ−r)v = 0.

What we have shown is that there exists a g ∈ G such that π(eI)π(g)v =0. Since v ∈ V I this implies that π(eIgI)v = 0. Now appealing to Lemma 9.13we get that v = 0. 2

37

Theorem 9.15 (Iwahori spherical representations) Let χ1, ..., χn be un-ramified characters of F ∗. Let π(χ) = π(χ1, ..., χn) be the corresponding prin-cipal series representation. Then any irreducible subquotient of π(χ) is Iwa-hori spherical and every Iwahori spherical representation arises in this man-ner.

Proof : Let π be any irreducible subquotient of π(χ). We know from The-orem 5.9 that the Jacquet module πU is non-zero. We also know from thecomputation of the Jacquet module of π(χ) in Theorem 5.4 that I0 acts triv-ially on π(χ)U since every χi is unramified and hence I0 acts trivially on πU ,i.e., (πU)I0 6= (0). Using Theorem 9.14 we get therefore that πI 6= (0), i.e., πis Iwahori spherical.

Now let π be any Iwahori spherical representation of G. We know thatπI0

U 6= (0) by the above theorem and hence there is an unramified characterχ of T such that HomT (πU , χ) 6= (0). Appealing to Frobenius reciprocityfinishes the proof. 2

Corollary 9.16 Let (π, V ) be any irreducible admissible representation ofG = GLn(F ). Then the dimension of space of fixed vectors under I is boundedabove by the order of the Weyl group W, i.e.,

dimC(V I) ≤ n!

Equivalently, the dimension of any simple module for the Iwahori Hecke al-gebra HI is bounded above by n!.

Proof : The corollary would follow if we show that the space of I fixed vectorsof an uramified principal series π(χ) is exactly n! which is the order of theWeyl group. This follows from Theorem 9.14 and Theorem 5.4. 2

10 Representations of local Galois Groups

The Galois group GF = Gal(F /F ) has distinguished normal subgroups IF ,the Inertia subgroup, and PF the wild Inertia subgroup which is containedin IF . The Inertia subgroup sits in the following exact sequence,

1 → IF → Gal(F /F ) → Gal(Fq/Fq) → 1.

Here the mapping from Gal(F /F ) to Gal(Fq/Fq) is given by the naturalaction of the Galois group of a local field on its residue field.

38

The Inertia group can be thought of as the Galois group of F over themaximal unramified extension F un of F . Let F t = ∪(d,q)=1F

un($1/dF ). The

field F t is known to be the maximal tamely ramified extension of F un (Anextension is called tamely ramified if the index of ramification is coprime tothe characteristic of the residue field.) We have,

Gal(F /F )/IF ' ZIF/PF '

∏` 6=p

Z×` .

One defines the Weil group WF of F to be the subgroup of Gal(F /F )whose image inside Gal(Fq/Fq) is an integral power of the Frobenius auto-morphism x → xq. Representations of the Weil group which are continuouson the inertia subgroup are exactly those representations of the Weil groupfor which the image of the Inertia subgroup is finite.

The Weil group is a dense subgroup of the Galois group hence an irre-ducible representation of the Galois group defines an irreducible representa-tion of the Weil group. It is easy to see that a representation of the Weilgroup can, after twisting by a character, be extended to a representation ofthe Galois group.

Local class field theory implies that the maximal abelian quotient of theWeil group of F is naturally isomorphic to F ∗, and hence 1 dimensionalrepresentations of WF are in bijective correspondence with characters ofGL1(F ) = F ∗. It is this statement of abelian class field theory which is gen-eralized by the local Langlands correspondence. However, there is a slightamount of change one needs to make, and instead of taking the Weil group,one needs to take what is called the Weil-Deligne group whose representa-tions are the same as representations of WF on a vector space V togetherwith a nilpotent endomorphism N such that

wNw−1 = |w|N

where |w| = q−i if the image of w in Gal(Fq/Fq) is the i-th power of theFrobenius.

One can identify representations of the Weil-Deligne group to representa-tions of WF ×SL2(C) via the Jacobson-Morozov theorem. It is usually mucheasier to work with WF × SL2(C) but the formulation with the nilpotentoperators appears more naturally in considerations of `-adic cohomology of

39

Shimura varieties where the nilpotent operator appears as the ‘monodromy’operator.

We end with the following important proposition for which we first notethat any field extension K of degree n of a local field F gives rise to aninclusion of the Weil group WK inside WF as a subgroup of index n. Sincecharacters of WK are, by local class field theory, identified to characters ofK∗, a character of K∗ gives by induction a representation of WF of dimensionn.

Proposition 10.1 If (n, q) = 1, then any irreducible representation of WF

of dimension n is induced from a character χ of K∗ for a field extension Kof degree n.

Proof : Since irreducible representations of the Weil group and Galois groupsare the same, perhaps after a twist, we will instead work with the Galoisgroup. Let ρ be an irreducible representation of the Galois group Gal(F /F )of dimension n > 1. We will prove that there exists an extension L of F ofdegree greater than 1, and an irreducible representation of Gal(F /L) whichinduces to ρ. This will complete the proof of the proposition by inductionon n.

The proof will be by contradiction. We will assume that ρ is not inducedfrom any proper subgroup.

We recall that there is a filtration on the Galois group PF ⊂ IF ⊂ GF =Gal(F /F ). Since PF is a pro-p group, all its irreducible representationsof dimension greater than 1 are powers of p. Since n is prime to p, thisimplies that there is a 1 dimensional representation of PF which appears in ρrestricted to PF . Since PF is a normal subgroup of the Galois group, it impliesthat the restriction of ρ to PF is a sum of characters. All the characters mustbe the same by Clifford theory. (Otherwise, the representation ρ is inducedfrom the stabilizer of any χ-isotypical component.) So, under the hypothesisthat ρ is not induced from any proper subgroup, PF acts via scalars on ρ.Since the exact sequence,

1 → PF → IF → IF/PF → 1

is a split exact sequence, letM be a subgroup in IF which goes isomorphicallyto IF/PF . Take a character of the abelian group M appearing in ρ. As PF

operates via scalars, the corresponding 1 dimensional space is invariant underIF . It follows that ρ restricted to IF contains a character. Again IF being

40

normal, ρ restricted to I is a sum of characters which must be all the sameunder the assumption that ρ is not induced from any proper subgroup. SinceG/IF is pro-cyclic, this is not possible. 2

11 The local Langlands conjecture for GLn

It is part of abelian classfield theory that for a local field F , the characters ofF ∗ can be identified to the characters of the Weil group WF of F . Langlandsvisualised a vast generalisation of this in the late 60’s to non-abelian repre-sentations of WF which is now a theorem due to M.Harris and R.Taylor, andanother proof was supplied shortly thereafter by G.Henniart.

The general conjecture of Langlands uses a slight variation of the Weilgroup, called the Deligne-Weil group, denoted W ′

F and defined to be W ′F =

WF × SL2(C).

Theorem 11.1 (Local Langlands Conjecture) There exists a natural bi-jective correspondence between irreducible admissible representations of GLn(F )and n-dimensional representations of the Weil-Deligne group W ′

F of F whichare semi-simple when restricted to WF and algebraic when restricted to SL2(C).The correspondence reduces to classfield theory for n = 1, and is equivariantunder twisting and taking duals.

The correspondence in the conjecture is called the local Langlands cor-respondence, and the n-dimensional representation of W ′

F associated to arepresentation π of GLn(F ) is called the Langlands parameter of π. TheLanglands correspondence is supposed to be natural in the sense that thereare L-functions and ε-factors attached to pairs of representations of W ′

F , andalso to pairs of representations of GLn(F ), characterising these representa-tions, and the correspondence is supposed to be the unique correspondencepreserving these. We will not define L-functions and ε-factors here, but referthe reader to the excellent survey article of Kudla [12] on the subject.

The results of Bernstein-Zelevinsky reviewed in section 7 and 8 reducethe Langlands correspondence between irreducible representations ofGLn(F )and representations of W ′

F to one between irreducible supercuspidal repre-sentations of GLn(F ) and irreducible representations of WF , as can be seenas follows.

An n-dimensional representation σ of the Weil-Deligne group W ′F of F

which is semi-simple when restricted to WF and algebraic when restricted to

41

SL2(C) is of the form

σ =i=r∑i=1

σi ⊗ Sp(mi)

where σi are irreducible representations of WF of dimension say ni, andSp(mi) is the unique mi dimensional irreducible representation of SL2(C).Assuming the Langlands correspondence between irreducible supercuspidalrepresentations of GLn(F ) and irreducible representations of WF , we haverepresentations πi of GLni

(F ) naturally associated to the representations σi

of dimension ni of WF . To the product mini, and a supercuspidal represen-tation πi of GLni

(F ), Theorem 7.5 associates an essentially square integrablerepresentation to the segment (πi(−mi−1

2), · · · , πi(

mi+12

)), which we denoteby Stmi

(πi). Now order the representations so that for i < j, the segment(πi(−mi−1

2), · · · , πi(

mi+12

)), does not precede (πj(−mj−1

2), · · · , πj(

mj+1

2)). By

Theorem 8.1, the parobolically induced representation

IndGP (Stm1(π1)⊗ · · · ⊗ Stmr(πr)),

admits a unique irreducible quotient which is the representation of GLn(F )associated by the Langlands correspondence to σ =

∑i=ri=1 σi ⊗ Sp(mi).

Example 11.2 The irreducible admissible representations π ofGL2 togetherwith the associated representation σπ of the Weil-Deligne group is as follows.

1. Principal series representation, induced from a pair of characters (χ1, χ2)of F ∗. These representations are irreducible if and only if χ1χ

−12 6= |·|±1.

For irreducible principal series representation, the associated Langlandsparameter is χ1⊕χ2, where χ1, χ2 are now being treated as charactersof WF .

2. Twists of Steinberg. The Langlands parameter of the Steinberg is thestandard 2 dimensional representation of SL2(C).

3. Twists of the trivial representation. The Langlands parameter of thetrivial representation is | · |1/2 ⊕ | · |−1/2.

4. The rest, which is exactly the set of supercuspidal representations ofGL2(F ). In odd residue characteristic, these representations can beconstructed by what is called the Weil representation, and the repre-sentation of GL2(F ) so constructed are parametrized by characters of

42

the invertible elements of quadratic field extensions of F . By proposi-tion 10.1, every 2 dimensional representation of WF , when the residuecharacteristic is odd, is also given by induction of such a character ona quadratic extension which is the Langlands parameter.

References

[1] D. Bump, Automorphic Forms and Representations, Cambridge studiesin Advanced Mathematics 55, (1997).

[2] I.N. Bernstein and A.V. Zelevinsky, Representation Theory of GL(n, F )where F is a non-Archimedea local field, Russian Math. Survey, 31:3,1-68, (1976).

[3] I.N. Bernstein and A.V. Zelevinsky, Induced representations of reduc-tive p-adic groups. I, Ann. Sci. Ecole Norm. Sup., (4) 10, no. 4, 441–472(1977).

[4] A. Borel, Admissible representations of a semi-simple group over a localfield with vectors fixed under am Iwahori subgroup, Inv. Math., 35, 233-259 (1976).

[5] C.J. Bushnell and P.C. Kutzko, The admissible dual of GL(N) viacompact open subgroups, Princeton university press, Princeton, (1993).

[6] H. Carayol, Representations cuspidales du groupe lineaire, Ann. Sci.Ecole. Norm. Sup., (4), Vol 17, 191-225 (1984).

[7] W. Casselman, An introduction to the theory of admissible representa-tions of reductive p-adic groups, Unpublished notes.

[8] I.M. Gelfand and S. Graev, Representations of a group pf matrices ofthe second order with elements from a locally compact field, and specialfunctions from a locally compact field, Russian Math Surveys, 18:4,29-100 (1963).

[9] J.A. Green,The characters of the finite general linear groups, Trans.Amer. Math. Soc. 80, 402–447 (1955).

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[10] G. Henniart, Une preuve simple des conjectures de Langlands pourGL(n) sur un corps p-adique. (French) [A simple proof of the Lang-lands conjectures for GL(n) over a p-adic field], Invent. Math., 139,no. 2, 439–455 (2000).

[11] N. Iwahori and H. Matsumoto, On some Bruhat decompositions and thestructure of the Hecke algebra rings of p-adic Chevalley groups, Publ.Math. I.H.E.S., 25, 5-48 (1965).

[12] H. Jacquet, Generic representations, in Non-Commutative harmonicanalysis, LNM 587, Springer Verlag, 91-101 (1977).

[13] Stephen S. Kudla, The local Langlands correspondence: the non-Archimedean case, Motives (Seattle, WA, 1991), 365–391, Proc. Sym-pos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI (1994).

[14] J.A. Shalika, The multiplicity one theorem for GLn, Ann. of Math., Vol100, 171-193 (1974).

[15] T.A. Springer, Linear Algebraic Groups, 2nd Edition, Progress inMathematics, Vol 9, Birkhauser, (1998).

[16] A.V. Zelevinsky, Induced representations of reductive p-adic groups II,Ann. Sci. Ecole Norm. Sup., (4) Vol 13, 165-210 (1980).

Dipendra PrasadHarish-Chandra Research Insititute, Chhatnag Road, Jhusi, Allahabad -211019, INDIA.e-mail: dprasad@mri.ernet.inA. RaghuramSchool of Mathematics, Tata Institute of Fundamental Research, Dr. HomiBhabha Road, Colaba, Mumbai - 400005, INDIA.e-mail : raghuram@math.tifr.res.in

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